Risks and Rewards

Money, money, money. Everyone wants more, but, alas, the second million won't do for you what the first one did, nor will you be as willing to take the same risks to get it.

It was Daniel Bernoulli, the 18th century Swiss mathematician, who wrote that people's enjoyment of any increase in wealth (or regret at any decrease) is "inversely proportionate to the quantity of goods previously possessed." The more dollars you have, the less you appreciate getting one more and the less you fear losing one.

What's important is the "utility" to you of the dollars you receive, and their utility drops off, often logarithmically, as you receive more of them. Gaining or losing $1 million means much more to most people than it does to Warren Buffett or Bill Gates. People consider not the dollar amount at stake in any investment or game, but the utility of the dollar amount for them.

Note that the declining average utility of money provides part of the rationale for progressive taxation and higher tax rates on greater wealth.

A less weighty illustration than progressive taxation is provided by a recent British study of the show "Who Wants To Be a Millionaire." It confirms that contestants behave as considerations of utility would suggest. Once they've reached a high rung on the winnings ladder, they more often quit while ahead rather than risk falling to a much lower level.

On the show and in general, people tend to be risk averse and usually choose the sure thing.

The St. Petersburg Paradox

Likewise, despite the equal expected values of the following alternatives, almost everyone offered the choice between i.) $100 million or ii.) a 1 percent chance at $10 billion will choose the sure $100 million. (Note that since 1 percent of $10 billion is $100 million, their expected values are equal.)

The notion of utility also resolves the famous St. Petersburg paradox. The paradox usually takes the form of a game requiring that you flip a coin repeatedly until a tail first appears. If a tail appears on the first flip, you win $2. If the first tail appears on the second flip, you win $4. If the first tail appears on the third flip, you win $8, and, in general, if the first tail appears on the Nth flip, you win 2^N dollars. How much would you be willing to pay to play this game?

In a sense you should be willing to pay any amount to play this game. To see why, recall that the probability of a sequence of independent events such as coin flips is obtained by multiplying the probabilities of each of the events. Thus the probability of getting the first tail, T, on the first flip is 1/2; of getting a head and then the first tail on the second flip, HT, is (1/2)^2 or 1/4; of getting the first tail on the third flip, HHT, is (1/2)^3 or 1/8, and so on.

Multiplying the probabilities for the various outcomes of the St. Petersburg game by the size of these outcomes and adding these products gives us the expected value of the game: ($2 x 1/2) + ($4 x 1/4) + ($8 x 1/8) + ($16 x 1/16) + … (2^N x [1/2]^N) + … Each of these products equals $1, there are infinitely many of them, and so their sum is infinite, and this is why it can be argued that you should be willing to pay any price to play this game. No matter how much you bet each time you play, you'll still come out way ahead on average.

But the failure of expected value to capture human intuitions becomes clear when you ask yourself why you'd be reluctant to pay even a measly $1,000 for the privilege of playing this game. That measly $1,000 is of more utility to you than are the billions of dollars that are only a remote possibility.

(Similar sorts of declining utility characterize other sorts of wealth. Someone who's publicity-rich will get a much smaller kick out of an article by him or her in a magazine, say, than someone who's never been published before. Someone who's traveled extensively all over the world will get less out of a few days in Florence than will someone else who's never left Torrance. And someone who's had sex with many partners will get … Well, you get the idea.)

The Ellsberg Paradox and the Sure Thing

To further discombobulate yourself, consider the so-called Ellsberg paradox, named after Daniel Ellsberg of Pentagon Papers fame. Imagine a large urn -- mathematicians like large urns almost as much as they like coins and dice -- containing 300 marbles, exactly 100 of them red, and the other 200 of them black and yellow in unknown proportions (i.e., from 0 to 200 black, the others yellow).

You're asked to choose option A or option B. Under A, you'll receive $1,000 if you pick a red marble from the urn, whereas under B, you'll receive $1,000 if you pick a black marble. Which option would you take?

You're also asked to choose between option C and option D. Under C, you'll receive $1,000 if you pick a red or a yellow marble from the urn, whereas under D you'll receive $1,000 if you pick a black or a yellow marble from the urn. Which option would you take here?

You'll prefer option A to option B exactly when you think picking a red marble from the urn is more probable than picking a black marble. Likewise, you'll prefer option C to option D exactly when you think picking a red or yellow marble from the urn is more probable than picking a black or yellow marble.

It would seem too that if you prefer option A to option B, you should also prefer C to D. And, if your prefer option D to option C, you should also prefer B to A.

The only problem is that this is not what people actually do. Most prefer Option A to B (presumably because they know for sure that there are 100 red marbles in the urn), but they prefer option D to option C (presumably because they know for sure that there are 200 black and yellow marbles in the urn). People are averse to uncertainty and choose the sure thing even when their behavior is inconsistent and violates the usual axioms of utility theory and subjective probability.

Most of us don't like risk and uncertainty. That's too bad, because there's no shortage of either.

-- Professor of mathematics at Temple University, John Allen Paulos is the author of best-selling books, including "Innumeracy" and "A Mathematician Plays the Stock Market." His "Who's Counting?" column on ABCNews.com appears the first weekend of every month.